This document discusses key concepts in reinforcement learning including agents, environments, states, actions, rewards, policies, episodes, returns, and discount factors. It defines an agent as a program that learns to make decisions through interactions with an environment. The agent receives rewards based on its actions and state transitions. The goal is for the agent to learn an optimal policy that maximizes long-term rewards through repeated trials and adjustments of its behavior.

1. Fundamentals of Reinforcement
Learning

2. Key elements of RL
• Agent:An agent is a software program that learns to make intelligent decisions.
• Environment:The environment is the world of the agent. The agent stays within
the environment.
• State:A state is a position or a moment in the environment that the agent can be in.
The state is usually denoted by s.
• Action:The agent interacts with the environment and moves from one state to
another by performing an action. The action is usually denoted by a.
• Reward:We learned that the agent interacts with an environment by performing an
action and moves from one state to another. Based on the action, the agent receives
a reward. A reward is nothing but a numerical value, say, +1 for a good action and
-1for a bad action. The reward is denoted by r.

3. RL agent in the grid world
Iteration 1 Iteration 2 Iteration 3
Note that these iterations are called episodes in RL terminology

4. Markov Decision Processes
• The Markov Decision Process (MDP) provides a mathematical framework for
solving the RL problem. Almost all RL problems can be modeled as an MDP.
• The Markov property : The Markov property states that the future depends only
on the present and not on the past.
• Markov chain : The Markov chain, also known as the Markov process, consists of
a sequence of states that strictly obey the Markov property; that is, the Markov
chain is the probabilistic model that solely depends on the current state to predict
the next state and not the previous states, that is, the future is conditionally
independent of the past.
• Moving from one state to another is called a transition, and its probability is called
a transition probability. We denote the transition probability by P(s′|s).
Markov Table
state diagram of a Markov chain
transition matrix

5. Markov Decision Processes
• The Markov Reward Process (MRP) is an extension of the Markov chain
with the reward function. That is, we learned that the Markov chain consists of
states and a transition probability. The MRP consists of states, a transition
probability, and also a reward function. A reward function tells us the reward
we obtain in each state. The reward function is usually denoted by R(s).
• the MRP consists of states s, a transition probability P(s′|s), and a reward
function R(s)
• The Markov Decision Process (MDP) is an extension of the MRP with
actions. The MDP consists of states, a transition probability, a reward function,
and also actions.
• An agent makes a decision (action) in the environment only based on the
current state the agent is in and not based on the past state. So, we can
formulate our environment as an MDP.
• States – A set of states present in the environment. Thus, in the grid world
environment, we have states A to I.
• Actions – A set of actions that our agent can perform in each state. An agent
performs an action and moves from one state to another. Thus, in the grid world
environment, the set of actions is up, down, left, and right.

6. Markov Decision Processes
• Transition probability – The transition probability is denoted by P(s′|s,a). It
implies the probability of moving from a state s to the next state 𝑠′ while
performing an action a.
• Reward function - The reward function is denoted by 𝑅(s,a,s′ ). It represents the
reward our agent obtains while transitioning from state s to state 𝑠′ while
performing an action a.
Transition probability of moving right from A to B
is expressed as P(B|A, right) = 1.0
Reward of moving right from A to B
expressed as R(A, right, B) = -1

7. Policy
• A policy defines the agent's behavior in an environment. The policy tells the agent
what action to perform in each state.
• To interact with the environment for the first time, we initialize a random policy,
that is, the random policy tells the agent to perform a random action in each state.
Thus, we can say that over a series of iterations, the agent will learn a good policy
that gives a positive reward.
• This good policy is called the optimal policy. The optimal policy is the policy that
gets the agent a good reward and helps the agent to achieve the goal.

8. Policy
• Deterministic Policy : A deterministic policy tells the agent to perform one
particular action in a state. Thus, the deterministic policy maps the state to one
particular action and is often denoted by 𝜇. Given a state s at a time t, a
deterministic policy tells the agent to perform one particular action a. It can be
expressed as: 𝑎t = 𝜇(st)
• For instance, consider our grid world example. Given state A, the deterministic
policy 𝜇𝜇 tells the agent to perform the action down. This can be expressed as:
𝜇(𝐴) = Down
• Stochastic Policy: a stochastic policy does not map a state directly to one
particular action; instead, it maps the state to a probability distribution over an
action space.
• is often denoted by 𝜋𝜋. Say we have a state s and action a at a time t, then we can
express the stochastic policy as: 𝑎t~ 𝜇(st)) Or 𝜋(𝑎t|st )

9. Episode
• The agent interacts with the environment by performing some actions, starting from
the initial state and reaches the final state. This agent-environment interaction
starting from the initial state until the final state is called an episode.
• An episode is also often called a trajectory (the path taken by the agent) and it is
denoted by 𝜏.
• state, action, and reward, starting from the initial state until the final state, that is,
(s0 , a0 , r0 , s1 , a1 , r1 ,…,sT).
• Episode 1
• Episode n

10. Episodic and continuous
• Episodic task: As the name suggests, an episodic task is one that has a terminal/
final state. That is, episodic tasks are tasks made up of episodes and thus they have
a terminal state.
• Continuous task: Unlike episodic tasks, continuous tasks do not contain any
episodes and so they don't have any terminal state.
Horizon
• Horizon is the time step until which the agent interacts with the environment. We
can classify the horizon into two categories:
• Finite horizon: If the agent-environment interaction stops at a particular time step,
then the horizon is called a finite horizon. starting from the initial state at time step
t = 0 and reaches the final state at time step T
• Infinite horizon: If the agent-environment interaction never stops, then it is called
an infinite horizon. the agent-environment interaction will never stop in a
continuous task and so it is considered an infinite horizon.

11. Return And Discount Factor
• A return can be defined as the sum of the rewards obtained by the agent in an
episode. The return is often denoted by R or G. Say the agent starts from the initial
state at time step t = 0 and reaches the final state at time step T, then the return
obtained by the agent is given as:𝑅(𝜏) = 𝑟0+𝑟1+𝑟2+...+𝑟T�
• consider the trajectory (episode) �
• The return of the trajectory is the sum of the rewards, that is, 𝑅(𝜏) =2+2+1+2=7
• we can define the return as a sum of rewards up to infinity: 𝑅(𝜏) = 𝑟0+𝑟1+𝑟2+...+𝑟 ∞
• But how can we maximize the return that just sums to infinity? We introduce a
new term called discount factor 𝛾𝛾 and rewrite our return as:
𝑅(𝜏) = 𝛾0 𝑟0 + 𝛾 1 𝑟1 + 𝛾2 𝑟2 + . . . +𝛾n𝑟∞

12. Return And Discount Factor
• The value of the discount factor ranges from 0 to 1. When we set the discount factor
to a small value (close to 0), it implies that we give more importance to immediate
rewards than to future rewards. When we set the discount factor to a high value
(close to 1), it implies that we give more importance to future rewards than to
immediate rewards.
• Small discount factor: Let's set the discount factor to a small value, say 0.2, that
is, let's set 𝛾=0.2, then we can write:
𝑅 = (𝛾)0 𝑟0 + (𝛾) 1 𝑟1 + (𝛾)2 𝑟2 + . . .
= (0.2)0 𝑟0 + (0.2) 1 𝑟1 + (0.2)2 𝑟2 + . . .
= (1)0 𝑟0 + (0.2) 1 𝑟1 + (0.04)2 𝑟2 + . . .
– At time step 0, the reward r0 is weighted by a discount factor of 1.
– At time step 1, the reward r1 is weighted by a heavily decreased discount factor of 0.2.
– At time step 2, the reward r2 is weighted by a heavily decreased discount factor of 0.04.
• The discount factor is heavily decreased for the subsequent time steps and more importance is
given to the immediate reward r0 than the rewards obtained at the future time steps.
• Thus, when we set the discount factor to a small value, we give more importance to the
immediate reward than future rewards.

13. Return And Discount Factor
• Large discount factor : Let's set the discount factor to a highvalue, say 0.9, that is, let's set 𝛾
=0.9, then we can write: 𝑅 = (𝛾)0 𝑟0 + (𝛾) 1 𝑟1 + (𝛾)2 𝑟2 + . . .
= (0.9)0 𝑟0 + (0.9) 1 𝑟1 + (0.9)2 𝑟2 + . . .
= (1)0 𝑟0 + (0.9) 1 𝑟1 + (0.81)2 𝑟2 + . . .
– At time step 0, the reward r0 is weighted by a discount factor of 1.
– At time step 1, the reward r1 is weighted by a slightly decreased discount factor of 0.9.
– At time step 2, the reward r2 is weighted by a slightly decreased discount factor of 0.81.
• The discount factor is decreased for subsequent time steps but unlike the previous case, the
discount factor is not decreased heavily.
• Thus, when we set the discount factor to a high value, we give more importance to future
rewards than the immediate reward.
What happens when we set the discount factor to 0?
• When we set the discount factor to 0, that is 𝛾=0, it implies that we consider only the
immediate reward 𝑟0 and not the reward obtained from the future time steps.
𝑅 = (𝛾)0 𝑟0 + (𝛾) 1 𝑟1 + (𝛾)2 𝑟2 + . . .
= (0)0 𝑟0 + (0) 1 𝑟1 + (0)2 𝑟2 + . . .
= (1)0 𝑟0 + (0)𝑟1 + (0)𝑟2 + . . .
= 𝑟0
• Thus, when we set the discount factor to 0, then the agent will never learn as it will consider
only the immediate reward 𝑟0

14. Return And Discount Factor
What happens when we set the discount factor to 1?
• When we set the discount factor to 1, that is 𝛾=1, it implies that we consider only
the future rewards.
𝑅 = (𝛾)0 𝑟0 + (𝛾) 1 𝑟1 + (𝛾)2 𝑟2 + . . .
= (1)0 𝑟0 + (1) 1 𝑟1 + (1)2 𝑟2 + . . .
= 𝑟0 + 𝑟1 + 𝑟2 + . . .
• Thus, when we set the discount factor to 1, then the agent will learn forever,
looking for all the future rewards, which may lead to infinity.
• Thus, we have learned that when we set the discount factor to 0, the agent will
never learn, considering only the immediate reward, and when we set the discount
factor to 1 the agent will learn forever, looking for the future rewards that lead to
infinity. So, the optimal value of the discount factor lies between 0.2 and 0.8.
• But the question is, why should we care about immediate and future rewards? We
give importance to immediate and future rewards depending on the tasks. In some
tasks, future rewards are more desirable than immediate rewards, and vice versa.

15. Value Function
• A Q function, also called the state-action value function, denotes the value of a
state-action pair. The value of a state-action pair is the return the agent would obtain
starting from state s and performing action a following policy 𝜋 .
• The value of a state or value function is usually denoted by V(s) and it can be
expressed a s: 𝑉 π(𝑠) = [𝑅(𝜏)| 𝑠0 = 𝑠]
where 𝑠0 = s implies that the starting state is s. The value of a state is called the
state value.
• we generate the trajectory 𝜏 following some policy 𝜋 in our grid world environment
• We learned that the value of a state is the return (sum of rewards) an agent would
obtain starting from that state following policy 𝜋. The preceding trajectory is
generated using policy 𝜋, thus we can say that the value of a state is the return (sum
of rewards) of the trajectory starting from that state.

16. Value Function
• The value of state A is the return of the trajectory starting from state A. Thus, V(A) = 1+1+ -1+1 = 2.
• The value of state D is the return of the trajectory starting from state D. Thus, V(D) = 1-1+1= 1.
• The value of state E is the return of the trajectory starting from state E. Thus, V(E) = -1+1 = 0.
• The value of state H is the return of the trajectory starting from state H. Thus, V(H) = 1.
• The value function or the value of state s can be defined as the expected return that the
agent would obtain starting from state s following policy 𝜋. It can be expressed as:
• First, we generate an episode 𝜏1 using our stochastic policy 𝜋,
• The value of state A is the return (sum of rewards) of the trajectory starting from state A.
Thus,𝑉(𝐴) = 𝑅(𝜏1) = 1 + 1 + 1 + 1 = 4 .

17. Value Function
• generate another episode 𝜏2 using the same given stochastic policy 𝜋
• The value of state A is the return (sum of rewards) of the trajectory from state A.
Thus, 𝑉(𝐴) = 𝑅(𝜏2) = −1 + 1 + 1 + 1 = 2
• The value of a state A can be obtained as:
=R(𝜏1) 𝜋(down|A)+ R(𝜏2) 𝜋(right|A)
=4(0.8)+2(0.2) =3.6

18. Value Function
• the value function depends on the policy, that is, the value of the state
varies based on the policy we choose. There can be many different value
functions according to different policies. The optimal value function V*(s)
yields the maximum value compared to all the other value functions. It can
be expressed as
• let's say we have two policies 𝜋1 and 𝜋2. Let the value of state s using policy 𝜋1 be
𝑉𝜋1(𝑠) = 13 and the value of state s using policy 𝜋2 be 𝑉𝜋2(𝑠) = 11.
• Then the optimal value of state s will be 𝑉∗(𝑠) = 13 as it is the maximum. The
policy that gives the maximum state value is called the optimal policy 𝜋∗. Thus, in
this case, 𝜋1 is the optimal policy as it gives the maximum state value
• We can view the value function in a table called a value table. Let's say we have
two states s0 and s1, then the value function can be represented as
State Value
S0 7
S1 11

19. Q Function
• A Q function, also called the state-action value function, denotes the value
of a state-action pair. The value of a state-action pair is the return the agent
would obtain starting from state s and performing action a following policy
𝜋 .
• The value of a state-action pair or Q function is usually denoted by Q(s,a)
and is known as the Q value or state-action value. It is expressed as:
𝑄𝜋(𝑠,a) = [𝑅(𝜏)|𝑠0 = 𝑠 ,a0 = a]
• difference between the value function and Q function is that in the value
function we compute the value of a state, whereas in the Q function we
compute the value of a state-action pair.
• Then the Q value will be the return of our trajectory starting from state A
and performing the action down:
𝑄𝜋(A,down) = [𝑅(𝜏)|𝑠0 = A ,a0 = down]
𝑄(A,down) =1+1+1+1=4

20. Q Function
• the expected return because the return is the random variable and it takes
different values with some probability. So, we can redefine our Q function
as:
• There can be many different Q functions according to different policies.
The optimal Q function is the one that has the maximum Q value over other
Q functions, and it can be expressed as:
• It is called a Q table.Let's say we have two states s0 and s1, and two
actions 0 and 1; then the Q function can be represented as follows:
State Actions Value
S0 0 9
S0 1 11
S1 0 17
S1 1 13
State Value
S0 1
S1 0
Q Table
Optimal Policy

21. Model-based and model-free learning
• Model-based learning: In model-based learning, an agent will have a
complete description of the environment. We know that the transition
probability tells us the probability of moving from state s to the next state 𝑠
𝑠′ by performing action a.
• The reward function tells us the reward we would obtain while moving
from state s to the next state 𝑠′ by performing action a.
• When the agent knows the model dynamics of its environment, that is,
when the agent knows the transition probability of its environment, then the
learning is called model-based learning.
• Model-free learning: Model-free learning is when the agent does not
know the model dynamics of its environment. That is, in model-free
learning, an agent tries to find the optimal policy without the model
dynamics.

22. Different types of environments
• Deterministic environment: In a deterministic environment, we are certain that
when an agent performs action a in state s, then it always reaches state 𝑠′.
• Stochastic environment: In a stochastic environment, we cannot say that by
performing action a in state s the agent always reaches state because there will
• Discrete environment: A discrete environment is one where the environment‘s
action space is discrete.
• Continuous environment: A continuous environment is one where the
environment's action space is continuous.
• Episodic environment: In an episodic environment, an agent's current action will
not affect future actions, and thus an episodic environment is also called a non-
sequential environment.
• Non-episodic environment: In a non-episodic environment, an agent's current
action will affect future actions, and thus a non-episodic environment is also called
a sequential environment.
• Single-agent environment: When our environment consists of only a single agent,
then it is called a single-agent environment.
• Multi-agent environment: When our environment consists of multiple agents, then
it is called a multi-agent environment.